Annual Reviews www.annualreviews.org/aronline Ann. Rev. Comput.Sci. 1987. 2 : 147~86 Copyright© 1987 by AnnualReviews Inc. All rights reserved NONMONOTONIC REASONING Raymond Reite? Department of Computer Science, University of Toronto, Toronto, Ontario M5S 1A4, Canada 1. INTRODUCTION If Artificial Intelligence (AI) researchers can agree on anything, it is that an intelligent artifact must be capable of reasoning about the world it inhabits. The artifact must possess various forms of knowledgeand beliefs about its world, and must use this information to infer further information about that world in order to makedecisions, plan and carry out actions, respond to other agents, etc. The technical problemfor AI is to characterize the patterns of reasoning required of such an intelligent artifact, and to realize them computationally. There is a wide range of such reasoning patterns necessary for intelligent behavior. Amongthese are: ¯ Probabilistic reasoning (e.g. Bundy1985; Nilsson 1986), in which prob- abilities are associated with different items of information. Reasoning requires, in part, computing appropriate probabilities for inferred information, based upon the probabilities of the information used to support the inference. ¯ Fuzzy reasoning (e.g. Zadeh 1981), designed to characterize vague concepts like "tall" or "old" and to assign degrees of vagueness to conclusions inferred using such concepts. ¯ Inductive reasoning (e.g. Michalski 1983), which is concerned with determining plausible generalizations from a finite number of obser- vations. ¯ Deductive reasoning, the concern of mathematical logic, which char- acterizes, amongother things, the axiomatic method in mathematics. This is far from a complete enumeration of humanreasoning patterns. The most recent addition to this list is nonmonotonicreasoning, the study Fellow of the Canadian Institute for AdvancedResearch. 147 8756-7016/87/1115-0147502.00 Annual Reviews www.annualreviews.org/aronline 148 REITER of which appears to be unique to AI. In order to convey an intuitive sense of what this is all about, it is first necessary to consider what has cometo be knownin AI as the knowledge representation problem. Because an agent must reason about something (its knowledge, beliefs), any consideration of the nature of reasoning requires a concomitant con- cern with how the agent represents its knowledgeand beliefs. The stance adopted by AI research on nonmonotonic reasoning is in agreement with the dominant view in AI on knowledge representation; the "knowledge content" of a reasoning programought to be represented by data structures interpretable as logical formulas of some kind. As Levesque (1986) puts it: For the structures to representknowledge, it mustbe possibleto interpret thempro- positionally,that is, as expressionsin a languagewith a truth theory.We should be able to pointto oneof themand say whatthe worldwould have to be like for it to be true. The province of nonmonotonicreasoning is the derivation of plausible (but not infallible) conclusions from a knowledgebase viewed abstractly as a set of formulas in a suitable logic. Anysuch conclusion is understood to be tentative; it mayhave to be retracted after new information has been added to the knowledge base. In what follows, I assumethe reader is logically literate, at least with respect to the fundamental ideas of first-order logic (with a smattering of second-order) and the familiar modal logics of necessity (e.g. $4 and $5). 2. MOTIVATION Nonmonotonicreasoning is a particular kind of plausible reasoning. Vir- tually every examplein AI that calls upon such reasoning fits the following pattern: Normally, A holds. Several paraphrases of this pattern are commonlyaccepted: Typically, A is the case. AssumeA by default. The remainder of this section is devoted to a number of examples of this pattern as it arises in various settings of special concern to AI. The ubiquity of this pattern is remarkable. Onceone learns to look for it, one discovers it virtually everywhere. Annual Reviews www.annualreviews.org/aronline NONMONOTONIC REASONING 149 2.1 The Canonical Example The standard example in AI of a nonmonotonic reasoning pattern has to do with flying birds. The sentence "Birds fly" is not synonymouswith "All birds fly" because there are exceptions. In fact, there are overwhelmingly manyexceptions--ostriches, penguins, Peking ducks, tar-coated birds, fledglings, etc. etc.--a seeminglyopen-ended list. Nevertheless, if told only about a particular bird, say Tweety, without being told anything else about it, we would be justified in assuming that Tweetycan fly, without knowing that it is not one of the exceptional birds. In other words, we treat Tweety as a typical or normalbird. Wecan represent the sentence "Birds fly" by instances of our patterns of plausible reasoning: "Normal, birds fly." "Typically, birds fly." "It x is a bird, then assumeby default that x flies." Wecan nowsee whythese are plausible reasoning patterns. Wewish to use them to conclude that Tweety can fly, but should we subsequently learn information to the contrary--say, that Tweety is a penguin--we would retract our earlier conclusion and conclude instead that Tweety cannot fly. Thus initially we jumpedto the conclusion or madethe default assumption that Tweety can fly. This, of course, is what makes our rule patterns plausible rather than deductive; they sanction assumptions rather than infallible conclusions. Notice also that there is another possible paraphrase of our reasoning pattern. In the case of Tweety the bird we were prepared to assume that Tweety can fly provided we knew of no information to the contrary, namelythat Tweetyis a penguin or an ostrich or the Maltese Falcon or .... So one possible reading of our pattern of plausible reasoning is: In the absence of information to the contrary, assume A. Whatis problematic here (as it is for notions like "typically" and "nor- mally") is defining precisely what one means by "absence of information to the contrary." A natural reading is something like "nothing is known that is inconsistent with the desired assumption A." As we shall see later, this consistency-based version of the pattern motivates several formal theories of nonmonotonicreasoning. Weshall also see that other intuitions are possible, leading to formalisms that apparently have little to do with consistency. 2.2 Databases In the theory of databases there is an explicit convention about the rep- resentation of negative information that appeals to a particular kind of Annual Reviews www.annualreviews.org/aronline 150 REITER default assumption. To see why negative information poses a problem, consider the simple example of a database for an airline flight schedule representing flight numbersand the city ipairs they connect. Wecertainly would not want to include in this database all flights and the city pairs they do not connect, which clearly would be an overwhelming amount of information. For example, Air Canada flight 103 does not connect London with Paris, or Toronto with Montreal, or MooseJaw with Athens, or .... There is far too muchne#ative information to represent explicity, and this will be true for any realistic database. Instead of explicitly representing such negative information, databases implicitly do so by appealing to the so-called closed worm assumption (Reiter 1978b), whichstates that all relevant positive information has been explicitly represented. If a positive fact is not explicitly present in the database, its negation is assumedto hold. For simple databases consisting of atomic facts only, e.g. relational databases, this approach to negative information is straightforward. In the case of deductive databases, however, the closed world assumption (CWA)is not so easy to formulate. It is no longer sufficient that a fact not be explicitly present in order to conjecture its negation; the fact maybe derivable. So in general we need a closed world rule that, for the flight schedule example, looks something like: If f is a flight and cl, c2 are cities, then in the absenceof informationto the contrary, assume -~ CONNECT(f,cl, c2). Here, by "absence of information to the contrary" we mean that CON- NECT(f,cl, c2) is not derivable using the database as premises. As shall see below, there are formal difficulties with this version of the CWA; but on an intuitive level the CWAconforms to the pattern of plausible reasoning we are considering in this section. Whenwe consider various proposed formalizations for nonmonotonic reasoning, below, we shall return to the question of the CWAsince it plays a dominant role in many approaches. 2.3 Diagnosis from First Principles There are two basic approaches in the AI literature to diagnostic reasoning. Under the first approach, which might be called the experiential approach, heuristic information plays a dominant role. The corresponding systems attempt to codify the rules of thumb, statistical intuitions, and past experience of humandiagnosticians considered experts in some par- ticular task domain. In particular, the structure or design of the object being diagnosed is only weakly represented, if at all. Successful diagnoses stem primarily from the codified experience of the humanexpert being Annual Reviews www.annualreviews.org/aronline NONMONOTONICREASONING 151 modeled rather than from detailed information about the object being diagnosed. This is the basis of so-called rule-based approachesto diagnosis, of which the MYCINsystem (Buchanan & Shortliffe 1984) is a notable example. Under